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Equilibrium equations for plane slender bars undergoing large shear deformations

Gleichgewichtsbedingungen ebener schlanker Stäbe bei großen Schubverformungen

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Summary

Equilibrium equations for plane slender bars are derived using the principles of Continuum Solid Mechanics. Large shear deformations are accounted for. The equations are derived in an explicit form, yielding upon introduction of a constitutive law (not given here) a system of differential equations to be integrated on the undeformed length. Within certain limitations, the equations, derived in an exact form, incorporate expressions for finite strain. The stress distribution along a deformed section is studied.

Zusammenfassung

Gleichgewichtsbedingungen ebener schlanker Stäbe werden unter Verwendung der Sätze der Kontinuumstechnik hergeleitet. Große Schubverformungen werden berücksichtigt. Die Gleichungen werden in expliziter Form angegeben, sie führen mit einem hier nicht angegebenen Werkstoffgesetz auf ein System von Differentialgleichungen. Mit gewissen Einschränkungen schließen diese in exakter Form angegebenen Gleichungen die für endliche Verformungen ein. Die Spannungsverteilung in einem verformten Abschnitt wird untersucht.

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Abbreviations

\(\vec r\) :

position vector of material point in undeformed state

\(\overrightarrow R\) :

position vector of material point in deformed state

\(\overrightarrow U\) :

displacement vector of material point

u :

displacement component inx-direction

w :

displacement component inz-direction

\(\vec e_i\) :

covariant base vector in deformed state (i=1,2)

G ij :

metric tensor in deformed state

g ij :

metric tensor in undeformed state

θ:

slope of geometrical axis (elastic line)

γ:

shear deformation angle

ψ:

rotation of plane section of bar

K :

curvature of geometrical axis

\(\overline K\) :

curvature of geometrical axis due to bending only

\(\vec t\) :

unit tangent vector to geometrical axis

\(\overrightarrow N\) :

unit normal vector to geometrical axis

η(ij) :

Lagrangian strain tensor

σij :

stress tensor

σ(ij) :

physical stress tensor

N :

normal force (parallel to geometrical axis) acting on plane section

S :

shear force (parallel to deformed section) acting on plane section

M :

bending moment

H :

horizontal external force acting on plane section

V :

vertical external force acting on plane section

\(\vec f\) :

vector of conservative body forces uniformly distributed along geometrical axis

f x :

component of\(\vec f\) inx-direction

f z :

component of\(\vec f\) inz-direction

\(\vec n\) :

unit normal vector to plane section

F i :

force component on distorted plane section

F (i) :

physical force component on distorted plane section

F ij :

component ofF i inj-direction

α:

inclination angle ofF 11 toF 1 (see Fig. 8)

References

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Authors and Affiliations

  1. Faculty of Civil Engineering, Technion-Israel Institute of Technology, Haifa, Israel

    A. Pisanty &  Y. Tene

Authors
  1. A. Pisanty
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  2. Y. Tene
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Pisanty, A., Tene, Y. Equilibrium equations for plane slender bars undergoing large shear deformations. Acta Mechanica 17, 263–275 (1973). https://doi.org/10.1007/BF01183760

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